Cell-specific models of hiPSC-CMs developed by the gradient-based parameter optimization method fitting two different action potential waveforms.

Parameter optimization (PO) methods to determine the ionic current composition of experimental cardiac action potential (AP) waveform have been developed using a computer model of cardiac membrane excitation. However, it was suggested that fitting a single AP record in the PO method was not always successful in providing a unique answer because of a shortage of information. We found that the PO method worked perfectly if the PO method was applied to a pair of a control AP and a model output AP in which a single ionic current out of six current species, such as IKr, ICaL, INa, IKs, IKur or IbNSC was partially blocked in silico. When the target was replaced by a pair of experimental control and IKr-blocked records of APs generated spontaneously in human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), the simultaneous fitting of the two waveforms by the PO method was hampered to some extent by the irregular slow fluctuations in the Vm recording and/or sporadic alteration in AP configurations in the hiPSC-CMs. This technical problem was largely removed by selecting stable segments of the records for the PO method. Moreover, the PO method was made fail-proof by running iteratively in identifying the optimized parameter set to reconstruct both the control and the IKr-blocked AP waveforms. In the lead potential analysis, the quantitative ionic mechanisms deduced from the optimized parameter set were totally consistent with the qualitative view of ionic mechanisms of AP so far described in physiological literature.

Ionic Mechanisms of Propagated Repolarization in a One-Dimensional Strand of Human Ventricular Myocyte Model.

Although repolarization has been suggested to propagate in cardiac tissue both theoretically and experimentally, it has been challenging to estimate how and to what extent the propagation of repolarization contributes to relaxation because repolarization only occurs in the course of membrane excitation in normal hearts. We established a mathematical model of a 1D strand of 600 myocytes stabilized at an equilibrium potential near the plateau potential level by introducing a sustained component of the late sodium current (INaL). By applying a hyperpolarizing stimulus to a small part of the strand, we succeeded in inducing repolarization which propagated along the strand at a velocity of 1~2 cm/s. The ionic mechanisms responsible for repolarization at the myocyte level, i.e., the deactivation of both the INaL and the L-type calcium current (ICaL), and the activation of the rapid component of delayed rectifier potassium current (IKr) and the inward rectifier potassium channel (IK1), were found to be important for the propagation of repolarization in the myocyte strand. Using an analogy with progressive activation of the sodium current (INa) in the propagation of excitation, regenerative activation of the predominant magnitude of IK1 makes the myocytes at the wave front start repolarization in succession through the electrical coupling via gap junction channels.

Gradient-based parameter optimization method to determine membrane ionic current composition in human induced pluripotent stem cell-derived cardiomyocytes.

Premature cardiac myocytes derived from human induced pluripotent stem cells (hiPSC-CMs) show heterogeneous action potentials (APs), probably due to different expression patterns of membrane ionic currents. We developed a method for determining expression patterns of functional channels in terms of whole-cell ionic conductance (Gx) using individual spontaneous AP configurations. It has been suggested that apparently identical AP configurations can be obtained using different sets of ionic currents in mathematical models of cardiac membrane excitation. If so, the inverse problem of Gx estimation might not be solved. We computationally tested the feasibility of the gradient-based optimization method. For a realistic examination, conventional 'cell-specific models' were prepared by superimposing the model output of AP on each experimental AP recorded by conventional manual adjustment of Gxs of the baseline model. Gxs of 4-6 major ionic currents of the 'cell-specific models' were randomized within a range of ± 5-15% and used as an initial parameter set for the gradient-based automatic Gxs recovery by decreasing the mean square error (MSE) between the target and model output. Plotting all data points of the MSE-Gx relationship during optimization revealed progressive convergence of the randomized population of Gxs to the original value of the cell-specific model with decreasing MSE. The absence of any other local minimum in the global search space was confirmed by mapping the MSE by randomizing Gxs over a range of 0.1-10 times the control. No additional local minimum MSE was obvious in the whole parameter space, in addition to the global minimum of MSE at the default model parameter.

Mechanisms Underlying Spontaneous Action Potential Generation Induced by Catecholamine in Pulmonary Vein Cardiomyocytes: A Simulation Study.

Cardiomyocytes and myocardial sleeves dissociated from pulmonary veins (PVs) potentially generate ectopic automaticity in response to noradrenaline (NA), and thereby trigger atrial fibrillation. We developed a mathematical model of rat PV cardiomyocytes (PVC) based on experimental data that incorporates the microscopic framework of the local control theory of Ca2+ release from the sarcoplasmic reticulum (SR), which can generate rhythmic Ca2+ release (limit cycle revealed by the bifurcation analysis) when total Ca2+ within the cell increased. Ca2+ overload in SR increased resting Ca2+ efflux through the type II inositol 1,4,5-trisphosphate (IP3) receptors (InsP3R) as well as ryanodine receptors (RyRs), which finally triggered massive Ca2+ release through activation of RyRs via local Ca2+ accumulation in the vicinity of RyRs. The new PVC model exhibited a resting potential of -68 mV. Under NA effects, repetitive Ca2+ release from SR triggered spontaneous action potentials (APs) by evoking transient depolarizations (TDs) through Na+/Ca2+ exchanger (APTDs). Marked and variable latencies initiating APTDs could be explained by the time courses of the α1- and ß1-adrenergic influence on the regulation of intracellular Ca2+ content and random occurrences of spontaneous TD activating the first APTD. Positive and negative feedback relations were clarified under APTD generation.

Regulation of the glucose supply from capillary to tissue examined by developing a capillary model.

A new glucose transport model relying upon diffusion and convection across the capillary membrane was developed, and supplemented with tissue space and lymph flow. The rate of glucose utilization (J util) in the tissue space was described as a saturation function of glucose concentration in the interstitial fluid (C glu,isf), and was varied by applying a scaling factor f to J max. With f = 0, the glucose diffusion ceased within ~20 min. While, with increasing f, the diffusion was accelerated through a decrease in C glu,isf, but the convective flux remained close to resting level. When the glucose supplying capacity of the capillary was measured with a criterion of J util /J max = 0.5, the capacity increased in proportion to the number of perfused capillaries. A consistent profile of declining C glu,isf along the capillary axis was observed at the criterion of 0.5 irrespective of the capillary number. Increasing blood flow scarcely improved the supplying capacity.

A new myofilament contraction model with ATP consumption for ventricular cell model.

A new contraction model of cardiac muscle was developed by combining previously described biochemical and biophysical models. The biochemical component of the new contraction model represents events in the presence of Ca2+-crossbridge attachment and power stroke following inorganic phosphate release, detachment evoked by the replacement of ADP by ATP, ATP hydrolysis, and recovery stroke. The biophysical component focuses on Ca2+ activation and force (F b) development assuming an equivalent crossbridge. The new model faithfully incorporates the major characteristics of the biochemical and biophysical models, such as F b activation by transient Ca2+ ([Ca2+]-F b), [Ca2+]-ATP hydrolysis relations, sarcomere length-F b, and F b recovery after jumps in length under the isometric mode and upon sarcomere shortening after a rapid release of mechanical load under the isotonic mode together with the load-velocity relationship. ATP consumption was obtained for all responses. When incorporated in a ventricular cell model, the contraction model was found to share approximately 60% of the total ATP usage in the cell model.

Modeling analysis of inositol 1,4,5-trisphosphate receptor-mediated Ca2+ mobilization under the control of glucagon-like peptide-1 in mouse pancreatic ß-cells.

Glucagon-like peptide-1 (GLP-1) is an intestinally derived blood glucose-lowering hormone that potentiates glucose-stimulated insulin secretion from pancreatic ß-cells. The secretagogue action of GLP-1 is explained, at least in part, by its ability to stimulate cAMP production so that cAMP may facilitate the release of Ca(2+) from inositol trisphosphate receptor (IP3R)-regulated Ca(2+) stores. However, a quantitative model has yet to be provided that explains the molecular mechanisms and dynamic processes linking GLP-1-stimulated cAMP production to Ca(2+) mobilization. Here, we performed simulation studies to investigate how GLP-1 alters the abilities of Ca(2+) and IP3 to act as coagonists at IP3R Ca(2+) release channels. A new dynamic model was constructed based on the Kaftan model, which demonstrates dual steady-state allosteric regulation of the IP3R by Ca(2+) and IP3. Data obtained from ß-cells were then analyzed to understand how GLP-1 facilitates IP3R-mediated Ca(2+) mobilization when UV flash photolysis is used to uncage Ca(2+) and IP3 intracellularly. When the dynamic model for IP3R activation was incorporated into a minimal cell model, the Ca(2+) transients and oscillations induced by GLP-1 were successfully reconstructed. Simulation studies indicated that transient and oscillatory responses to GLP-1 were produced by sequential positive and negative feedback regulation due to fast activation and slow inhibition of the IP3R by Ca(2+). The slow rate of Ca(2+)-dependent inhibition was revealed to provide a remarkable contribution to the time course of the decay of cytosolic Ca(2+) transients. It also served to drive and pace Ca(2+) oscillations that are significant when evaluating how GLP-1 stimulates insulin secretion.

Mechanisms underlying the volume regulation of interstitial fluid by capillaries: a simulation study.

BACKGROUND: Control of the extracellular fluid volume is one of the most indispensable issues for homeostasis of the internal milieu. However, complex interdependence of the pressures involved in determination of fluid exchange makes it difficult to predict a steady-state tissue volume under various physiological conditions without mathematical approaches. METHODS: Here, we developed a capillary model based on the Starling's principle, which allowed us to clarify the mechanisms of the interstitial-fluid volume regulation. Three well known safety factors against edema: (1) low tissue compliance in negative pressure ranges; (2) lymphatic flow driven by the tissue pressure; and (3) protein washout by the lymph, were incorporated into the model in sequence. RESULTS: An increase in blood pressure at the venous end of the capillary induced an interstitial-fluid volume increase, which, in turn, reduced negative tissue pressure to prevent edema. The lymphatic flow alleviated the edema by both carrying fluid away from the tissue and decreasing the colloidal osmotic pressure. From the model incorporating all three factors, we found that the interstitial-fluid volume changed quickly after the blood pressure change, and that the protein movement towards a certain equilibrium point followed the volume change. CONCLUSION: Mathematical analyses revealed that the system of the capillary is stable near the equilibrium point at steady state and normal physiological capillary pressure. The time course of the tissue-volume change was determined by two kinetic mechanisms: rapid fluid exchange and slow protein fluxes.

A human ventricular myocyte model with a refined representation of excitation-contraction coupling.

Cardiac Ca(2+)-induced Ca(2+) release (CICR) occurs by a regenerative activation of ryanodine receptors (RyRs) within each Ca(2+)-releasing unit, triggered by the activation of L-type Ca(2+) channels (LCCs). CICR is then terminated, most probably by depletion of Ca(2+) in the junctional sarcoplasmic reticulum (SR). Hinch et al. previously developed a tightly coupled LCC-RyR mathematical model, known as the Hinch model, that enables simulations to deal with a variety of functional states of whole-cell populations of a Ca(2+)-releasing unit using a personal computer. In this study, we developed a membrane excitation-contraction model of the human ventricular myocyte, which we call the human ventricular cell (HuVEC) model. This model is a hybrid of the most recent HuVEC models and the Hinch model. We modified the Hinch model to reproduce the regenerative activation and termination of CICR. In particular, we removed the inactivated RyR state and separated the single step of RyR activation by LCCs into triggering and regenerative steps. More importantly, we included the experimental measurement of a transient rise in Ca(2+) concentrations ([Ca(2+)], 10-15 µM) during CICR in the vicinity of Ca(2+)-releasing sites, and thereby calculated the effects of the local Ca(2+) gradient on CICR as well as membrane excitation. This HuVEC model successfully reconstructed both membrane excitation and key properties of CICR. The time course of CICR evoked by an action potential was accounted for by autonomous changes in an instantaneous equilibrium open probability of couplons. This autonomous time course was driven by a core feedback loop including the pivotal local [Ca(2+)], influenced by a time-dependent decay in the SR Ca(2+) content during CICR.

Steady-state solutions of cell volume in a cardiac myocyte model elaborated for membrane excitation, ion homeostasis and Ca2+ dynamics.

The cell volume continuously changes in response to varying physiological conditions, and mechanisms underlying volume regulation have been investigated in both experimental and theoretical studies. Here, general formulations concerning cell volume change are presented in the context of developing a comprehensive cell model which takes Ca(2+) dynamics into account. Explicit formulas for charge conservation and steady-state volumes of the cytosol and endoplasmic reticulum (ER) are derived in terms of membrane potential, amount of ions, Ca(2+)-bound buffer molecules, and initial cellular conditions. The formulations were applied to a ventricular myocyte model which has plasma-membrane Ca(2+) currents with dynamic gating mechanisms, Ca(2+)-buffering reactions with diffusive and non-diffusive buffer proteins, and Ca(2+) uptake into or release from the sarcoplasmic reticulum (SR) accompanied by compensatory cationic or anionic currents through the SR membrane. Time-dependent volume changes in cardiac myocytes induced by varying extracellular osmolarity or by action potential generation were successfully simulated by the novel formulations. Through application of bifurcation analysis, the existence and uniqueness of steady-state solutions of the cell volume were validated, and contributions of individual ion channels and transporters to the steady-state volume were systematically analyzed. The new formulas are consistent with previous fundamental theory derived from simple models of minimum compositions. The new formulations may be useful for examination of the relationship between cell function and volume change in other cell types.

Analyzing electrical activities of pancreatic ß cells using mathematical models.

Bursts of repetitive action potentials are closely related to the regulation of glucose-induced insulin secretion in pancreatic ß cells. Mathematical studies with simple ß-cell models have established the central principle that the burst-interburst events are generated by the interaction between fast membrane excitation and slow cytosolic components. Recently, a number of detailed models have been developed to simulate more realistic ß cell activity based on expanded findings on biophysical characteristics of cellular components. However, their complex structures hinder our intuitive understanding of the underlying mechanisms, and it is becoming more difficult to dissect the role of a specific component out of the complex network. We have recently developed a new detailed model by incorporating most of ion channels and transporters recorded experimentally (the Cha-Noma model), yet the model satisfies the charge conservation law and reversible responses to physiological stimuli. Here, we review the mechanisms underlying bursting activity by applying mathematical analysis tools to representative simple and detailed models. These analyses include time-based simulation, bifurcation analysis and lead potential analysis. In addition, we introduce a new steady-state I-V (ssI-V) curve analysis. We also discuss differences in electrical signals recorded from isolated single cells or from cells maintaining electrical connections within multi-cell preparations. Towards this end, we perform simulations with our detailed pancreatic ß-cell model.

Systems analysis of GLP-1 receptor signaling in pancreatic ß-cells.

Glucagon-like peptide-1 (GLP-1) elevates intracellular concentration of cAMP ([cAMP]) and facilitates glucose-dependent insulin secretion in pancreatic ß-cells. There has been much evidence to suggest that multiple key players such as the GLP-1 receptor, G(s) protein, adenylate cyclase (AC), phosphodiesterase (PDE), and intracellular Ca(2+) concentration ([Ca(2+)]) are involved in the regulation of [cAMP]. However, because of complex interactions among these signaling factors, the kinetics of the reaction cascade as well as the activities of ACs and PDEs have not been determined in pancreatic ß-cells. We have constructed a minimal mathematical model of GLP-1 receptor signal transduction based on experimental findings obtained mostly in ß-cells and insulinoma cell lines. By fitting this theoretical reaction scheme to key experimental records of the GLP-1 response, the parameters determining individual reaction steps were estimated. The model reconstructed satisfactorily the dynamic changes in [cAMP] and predicted the activities of cAMP effectors, protein kinase A (PKA), and cAMP-regulated guanine nucleotide exchange factor [cAMP-GEF or exchange protein directly activated by cAMP (Epac)] during GLP-1 stimulation. The simulations also predicted the presence of two sequential desensitization steps of the GLP1 receptor that occur with fast and very slow reaction rates. The cross talk between glucose- and GLP-1-dependent signal cascades for cAMP synthesis was well reconstructed by integrating the direct regulation of AC and PDE by [Ca(2+)]. To examine robustness of the signaling system in controlling [cAMP], magnitudes of AC and PDE activities were compared in the presence or absence of GLP-1 and/or the PDE inhibitor IBMX.(1).

Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic ß cells.

In our companion paper, the physiological functions of pancreatic ß cells were analyzed with a new ß-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst-interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca(2+)], and [Na(+)]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic ß cells.

Ionic mechanisms and Ca2+ dynamics underlying the glucose response of pancreatic ß cells: a simulation study.

To clarify the mechanisms underlying the pancreatic ß-cell response to varying glucose concentrations ([G]), electrophysiological findings were integrated into a mathematical cell model. The Ca(2+) dynamics of the endoplasmic reticulum (ER) were also improved. The model was validated by demonstrating quiescent potential, burst-interburst electrical events accompanied by Ca(2+) transients, and continuous firing of action potentials over [G] ranges of 0-6, 7-18, and >19 mM, respectively. These responses to glucose were completely reversible. The action potential, input impedance, and Ca(2+) transients were in good agreement with experimental measurements. The ionic mechanisms underlying the burst-interburst rhythm were investigated by lead potential analysis, which quantified the contributions of individual current components. This analysis demonstrated that slow potential changes during the interburst period were attributable to modifications of ion channels or transporters by intracellular ions and/or metabolites to different degrees depending on [G]. The predominant role of adenosine triphosphate-sensitive K(+) current in switching on and off the repetitive firing of action potentials at 8 mM [G] was taken over at a higher [G] by Ca(2+)- or Na(+)-dependent currents, which were generated by the plasma membrane Ca(2+) pump, Na(+)/K(+) pump, Na(+)/Ca(2+) exchanger, and TRPM channel. Accumulation and release of Ca(2+) by the ER also had a strong influence on the slow electrical rhythm. We conclude that the present mathematical model is useful for quantifying the role of individual functional components in the whole cell responses based on experimental findings.

Minor contribution of cytosolic Ca2+ transients to the pacemaker rhythm in guinea pig sinoatrial node cells.

The question of the extent to which cytosolic Ca(2+) affects sinoatrial node pacemaker activity has been discussed for decades. We examined this issue by analyzing two mathematical pacemaker models, based on the "Ca(2+) clock" (C) and "membrane clock" (M) hypotheses, together with patch-clamp experiments in isolated guinea pig sinoatrial node cells. By applying lead potential analysis to the models, the C mechanism, which is dependent on potentiation of Na(+)/Ca(2+) exchange current via spontaneous Ca(2+) release from the sarcoplasmic reticulum (SR) during diastole, was found to overlap M mechanisms in the C model. Rapid suppression of pacemaker rhythm was observed in the C model by chelating intracellular Ca(2+), whereas the M model was unaffected. Experimental rupturing of the perforated-patch membrane to allow rapid equilibration of the cytosol with 10 mM BAPTA pipette solution, however, failed to decrease the rate of spontaneous action potential within ∼30 s, whereas contraction ceased within ∼3 s. The spontaneous rhythm also remained intact within a few minutes when SR Ca(2+) dynamics were acutely disrupted using high doses of SR blockers. These experimental results suggested that rapid disruption of normal Ca(2+) dynamics would not markedly affect spontaneous activity. Experimental prolongation of the action potentials, as well as slowing of the Ca(2+)-mediated inactivation of the L-type Ca(2+) currents induced by BAPTA, were well explained by assuming Ca(2+) chelation, even in the proximity of the channel pore in addition to the bulk cytosol in the M model. Taken together, the experimental and model findings strongly suggest that the C mechanism explicitly described by the C model can hardly be applied to guinea pig sinoatrial node cells. The possible involvement of L-type Ca(2+) current rundown induced secondarily through inhibition of Ca(2+)/calmodulin kinase II and/or Ca(2+)-stimulated adenylyl cyclase was discussed as underlying the disruption of spontaneous activity after prolonged intracellular Ca(2+) concentration reduction for >5 min.

Magnesium gating of cardiac gap junction channels.

We aimed to study kinetics of modulation by intracellular Mg(2+) of cardiac gap junction (Mg(2+) gate). Paired myocytes of guinea-pig ventricle were superfused with solutions containing various concentrations of Mg(2+). In order to rapidly apply Mg(2+) to one aspect of the gap junction, the non-junctional membrane of one of the pair was perforated at nearly the connecting site by pulses of nitrogen laser beam. The gap junction conductance (G(j)) was measured by clamping the membrane potential of the other cell using two-electrode voltage clamp method. The laser perforation immediately increased G(j), followed by slow G(j) change with time constant of 3.5 s at 10 mM Mg(2+). Mg(2+) more than 1.0 mM attenuated dose-dependently the gap junction conductance and lower Mg(2+) (0.6 mM) increased G(j) with a Hill coefficient of 3.4 and a half-maximum effective concentration of 0.6 mM. The time course of G(j) changes was fitted by single exponential function, and the relationship between the reciprocal of time constant and Mg(2+) concentration was almost linear. Based on the experimental data, a mathematical model of Mg(2+) gate with one open state and three closed states well reproduced experimental results. One-dimensional cable model of thirty ventricular myocytes connected to the Mg(2+) gate model suggested a pivotal role of the Mg(2+) gate of gap junction under pathological conditions.

Characterization of the cardiac Na+/K+ pump by development of a comprehensive and mechanistic model.

A large amount of experimental data on the characteristics of the cardiac Na(+)/K(+) pump have been accumulated, but it remains difficult to predict the quantitative contribution of the pump in an intact cell because most measurements have been made under non-physiological conditions. To extrapolate the experimental findings to intact cells, we have developed a comprehensive Na(+)/K(+) pump model based on the thermodynamic framework (Smith and Crampin, 2004) of the Post-Albers reaction cycle combined with access channel mechanisms. The new model explains a variety of experimental results for the Na(+)/K(+) pump current (I(NaK)), including the dependency on the concentrations of Na(+) and K(+), the membrane potential and the free energy of ATP hydrolysis. The model demonstrates that both the apparent affinity and the slope of the substrate-I(NaK) relationship measured experimentally are affected by the composition of ions in the extra- and intracellular solutions, indirectly through alteration in the probability distribution of individual enzyme intermediates. By considering the voltage dependence in the Na(+)- and K(+)-binding steps, the experimental voltage-I(NaK) relationship could be reconstructed with application of experimental ionic compositions in the model, and the view of voltage-dependent K(+) binding was supported. Re-evaluation of charge movements accompanying Na(+) and K(+) translocations gave a reasonable number for the site density of the Na(+)/K(+) pump on the membrane. The new model is relevant for simulation of cellular functions under various interventions, such as depression of energy metabolism.

A novel method to quantify contribution of channels and transporters to membrane potential dynamics.

The action potential, once triggered in ventricular or atrial myocytes, automatically proceeds on its time course or is generated spontaneously in sinoatrial node pacemaker cells. It is induced by complex interactions among such cellular components as ion channels, transporters, intracellular ion concentrations, and signaling molecules. We have developed what is, to our knowledge, a new method using a mathematical model to quantify the contribution of each cellular component to the automatic time courses of the action potential. In this method, an equilibrium value, which the membrane potential is approaching at a given moment, is calculated along the time course of the membrane potential. The calculation itself is based on the time-varying conductance and the reversal potentials of individual ion channels and electrogenic ion transporters. Since the equilibrium potential moves in advance of the membrane potential change, we refer to it as the lead potential, V(L). The contribution of an individual current was successfully quantified by comparing dV(L)/dt before and after fixing the time-dependent change of a component of interest, such as the variations in the open probability of a channel or the turnover rate of an ion transporter. In addition to the action potential, the lead-potential analysis should also be applicable in all types of membrane excitation in many different kinds of cells.

A model of Na+/H+ exchanger and its central role in regulation of pH and Na+ in cardiac myocytes.

A new kinetic model of the Na(+)/H(+) exchanger (NHE) was developed by fitting a variety of major experimental findings, such as ion-dependencies, forward/reverse mode, and the turnover rate. The role of NHE in ion homeostasis was examined by implementing the NHE model in a minimum cell model including intracellular pH buffer, Na(+)/K(+) pump, background H(+), and Na(+) fluxes. This minimum cell model was validated by reconstructing recovery of pH(i) from acidification, accompanying transient increase in [Na(+)](i) due to NHE activity. Based on this cell model, steady-state relationships among pH(i), [Na(+)](I), and [Ca(2+)](i) were quantitatively determined, and thereby the critical level of acidosis for cell survival was predicted. The acidification reported during partial blockade of the Na(+)/K(+) pump was not attributed to a dissipation of the Na(+) gradient across the membrane, but to an increase in indirect H(+) production. This NHE model, though not adapted to the dimeric behavioral aspects of NHE, can provide a strong clue to quantitative prediction of degree of acidification and accompanying disturbance of ion homeostasis under various pathophysiological conditions.